The present invention relates to high current, high energy electron beams. More specifically, the present invention relates to a modified betatron which has beam self forces that are inward in the polodial plane.
There is a great interest in the development of high current, high energy electron beams. The principal technique to accelerate electron rings, which comprise the electron beam, is a conventional betatron. See, for instance U.S. Pat. No. 4,392,111, by Rostaker, 1983. However, this technique is limited by the fact that the outward self forces of the electron beam must be smaller than the externally imposed focusing forces of the betatron. These focusing forces are controlled by the gradient of the vertical(z) field, the field parallel to the axis of the betatron. The focusing is parameterized by the field index ##EQU1## For 0&lt;.eta.&lt;1, the focusing forces are inward in both the radial and vertical directions in the poloidal plane. In practice, this limits the current to several hundred amps. Another scheme is to use a plasma betatron. See, Taggert et al., "Successful Betatron Acceleration of Kiloampere Electron Rings in RECE-Christa", Physical Review Letters, Vol. 52, No. 18, p. 1601-1604, April 1984, for a more complete description of a plasma betatron. Here a runaway tokamak discharge is produced which can have a current of many kiloamps and a voltage of 10 MeV or more. However, the beam has a large energy spread with no way to extract it from either the plasma or the magnetic field. Another scheme is a modified betatron which uses a toroidal magnetic field to overcome the lack of external field focusing. See U.S. Pat. No. 4,481,475, by Kapetanakos and Sprangle, 1984, for a more detailed description of the modified betatron.
However, the disadvantages of the modified betatron are threefold. First of all, one must inject the beam across toroidal field lines in order to have a beam centered in the liner of the betatron. The current scheme of Kapetanakos et al., Phys, Rev. Lett. 49, 741 (1982) proposes to shoot the beam into the toroidal vacuum chamber near the liner. The drift due to the focusing fields and image fields causes the beam to drift in the poloidal plane around the liner. In one toroidal transit (about 20 nsec) it should drift enough to miss the injector. In one poloidal drift time (several hundred nanoseconds), external fields can be changed to bring the beam slightly in from the liner so that it misses the injector again and henceforth. On a longer time scale, wall resistivity causes the beam (if the current is sufficiently low) to drift inward. This occurs because the liner has finite conductivity. The effect of this finite conductivity is to cause a drag force on the beam which causes the beam to spiral either in or out, depending on the beam current.
However there is a significant range of beam currents for which wall resistivity causes the beam to drift outward if it is near the liner, but inward if it is near the center. Since it is unlikely that the beam can reverse drift directions on the way in, the injection scheme of Kapetanakos et al., supra. appears to be viable only for fairly low beam currents. For higher beam currents still, the beam will drift outward no matter what its position is in the poloidal plane.
A second possible difficulty concerning the modified betatron is the diamagnetic to paramagnetic transition. See W. M. Manheiner and J. M. Finn, Particle Accel. 14, 29 (1983); J. M. Finn and W. M. Manheimer, Phys. Fluids 26, 3400 (1983). Depending on whether the net self force is outward or inward in the poloidal plane, the net electron drift velocity in the poloidal plane is in the diamagnetic or paramagnetic direction. A diamagnetic drift velocity means that the electron poloidal velocity crossed with the toroidal magnetic field (right hand rule) produces an inward force in the poloidal plane. A paramagnetic drift means that this force is outward in the poloidal plane. At low energy, where the outward self fields are stronger than the focusing forces, the beam must have an additional inward force to maintain equilibrium. At high energy, where the self forces are weaker than the focusing forces, the beam must have an additional outward force to maintain equilibrium. These forces are provided by the poloidal (diamagnetic or paramagnetic) drift velocity times the toroidal magnetic field. Thus as the high current beam accelerates, it makes a transition from diamagnetic to paramagnetic current flow. It has been shown that subject only to the constraint that the acceleration time .tau..sub.a .apprxeq.10.sup.-3 sec is very long compared to the drift time, .tau..sub.D .gtoreq.10.sup.-7 sec, this transition must suddenly change the topology of the beam orbits in the poloidal plane. Whether the beam can survive such a sudden, violent perturbation is an open question.
Finally, although the focusing fields in the modified betatron stabilize the l=1 resistive wall instability, l=2 modes are still unstable and pose a real threat to beam confinement in the modified betatron. See R. G. Kleva, E. Ott and P. Sprangle, Phys. Fluids 26, 2689 (1983). The l=1 modes causes a displacement of the beam center in the poloidal plane. The l=2 modes causes a distortion of the shape of the beam in the poloidal plane from circular to elliptical.
The three fundamental issues identified regarding the high current modified betatron operating with a vacuum background: beam injection, the diamagnetic to paramagnetic transition, and the l=2 resistive wall instability will now be more fully discussed.
One of the important issues for the modified betatron is injecting the beam. The present thinking for the Naval Research Laboratory modified betatron experiment is described in Kapetanakos, et al., supra. The beam is injected near the liner and drifts around the edge of the liner through a combination of drift paths generated by the focusing fields (field index) of the betatron and image forces in the vacuum chamber wall due to the beam. The former is directed inward in the poloidal plane, the latter, outward. At high beam current the latter dominates and the beam rotates in a counterclockwise direction as in FIG. 3 of Kapetanakos et al., supra. for the case of a ten kilo Amp beam. If the combination of forces is large enough, the drift velocity will be great enough so that after one toroidal revolution, the beam will be displaced in the poloidal plane by a large enough distance that it misses the injector. Then, since it has many more toroidal transits before it would hit the injector again, macroscopic fields could change sufficiently to bring the beam into the center.
One potential problem with this scheme, is that for higher current beams, the net poloidal force on the beam is outward near the liner, but inward when the beam is at the center. Thus, as the beam continues to spiral in the poloidal plane, at some point it must reverse direction. To see this more quantitatively, if the field index of the beam is 1/2, which gives rise to optimal confinement in the radial and vertical direction, the focusing field produces an inward poloidal force on a charge q of ##EQU2## where R.sub.o is the major radius of the equilibrium orbit, B.sub.z is the vertical field and .rho. is the displacement of the beam from the equilibrium orbit in the poloidal plane .rho..sup.2 =(R-R.sub.o).sup.2 +z.sup.2. The image electric force for a cylindrical system is given by ##EQU3## where a is the minor radius of the liner.
The actual force is canceled in part by magnetic forces and also by any fractional neutralization f. The fractional neutralization describes the fact that positive ions, with a density of f times the beam density could also be confined by the beam. If the beam is near the wall (.rho..ltoreq.a) the net poloidal drift velocity is given by ##EQU4## where .delta. is the distance from the beam center to the liner and .gamma. is the electron energy divided by the electron rest energy. Since the beam enters the toroidal liner right near the outer edge of the liner, .delta. is roughly equal to the beam radius .rho..sub.b.
If the beam is near the center (so .rho.&lt;a), the poloidal drift is given by ##EQU5## For the case of a vacuum modified betatron, the current can be classified as being in one of three ranges:
I. High Current ##EQU6## II. Intermediate Current ##EQU7## III. Low Current ##EQU8##
In the high current regime, the forces on the beam in the poloidal plane are outward and the beam always rotates in the counterclockwise direction. In the intermediate regime, the forces are outward when the beam is near the wall, but inward when the beam is near the center. Thus in this current regime, the beam must reverse its direction of rotation before it gets to the center. Also, at some radius between the center and the wall, the beam will have zero poloidal drift velocity. It seems likely that some time after injection, an intermediate current beam will stagnate around this point and gradually fill the chamber. In the low current regime, the inward focusing forces always dominate and the beam rotation is clockwise.
It is also worth noting that if the liner is resistive, the beam will spiral inward if the net force is inward and visa versa. Thus a resistive wall can only trap a class III low current beam. It is possible that an intermediate current beam can be trapped if it can be brought sufficiently near the center that the net forces are inward. For the parameters of the NRL modified betatron experiment, (see Sprangle et al., supra; Kapentanakas et al., supra, B.sub..theta. =2.times.10.sup.3, .gamma.=4, R.sub.o =10.sup.2, a=15, .delta.=2, B.sub.z =140, the lower and upper currents of the intermediate range are 3.2.times.10.sup.3 A and 1.2.times.10.sup.4 A. Thus the maximum current which can be trapped by wall resistivity in a vacuum modified betatron is about 3 kA. Actually however, the maximum current is less because at the 3.2 kA level the poloidal drift is zero, so the beam will strike the injector after one toroidal transit. Note that here a cylindrical system is assumed.
Referring to the diamagnetic to paramagnetic transition, once the beam has been injected and is centered in the modified betatron, the question then is about the individual particle orbits in the beam. Each particle feels an inward force due to the focusing fields and an outward force due to the self fields. If the latter dominates, the particle has an F.times.B drift in the counterclockwise direction, analogous to the counterclockwise whole beam drift for an outward image force discussed above. Then the J (poloidal) B (toroidal) force is inward. In this case, the beam is said to be diamagnetic. This is analogous to the terminology in plasma physics, where the poloidal current is diamagnetic if it gives an inward force. On the other hand, if the focusing force dominates, the F.times.B drift is clockwise and the beam is said to be paramagnetic.
If the beam has uniform density and radius .rho..sub.b, the outward force is the electrostatic force canceled by the magnetic force and fractional charge neutralization. A test charge q at .rho.=.rho..sub.b feels an outward force ##EQU9## The inward focusing force is given by ##EQU10## so the condition for a paramagnetic beam in a vacuum modified betatron is ##EQU11## Note that a high current beam is generally diamagnetic. However as it accelerates, the left hand side becomes smaller as .gamma. increases, and the right hand side becomes larger because B.sub.z is proportional to .gamma.. Thus for a high current beam which starts out diamagnetic, as it accelerates it ultimately makes a transition and becomes paramagnetic.
One might think this simply means that the poloidal rotation of the particle stops and changes direction. Actually the situation is considerably more complex, and also worse from the point of view of operation of the modified betatron. In a recent series of papers, see Manheimer, supra; Finn, supra; J. M. Grossman, J. M. Finn and W. M. Manheimer, Phys. Fluids, to be published, it has been shown that subject only to the constraint that the acceleration time is very long compared to the drift time, an approximation well-satisfied in the NRL modified betatron (but not satisfied at all in particle simulations of the device), the diamagnetic to paramagnetic transition necessarily results in a change of topology of the beam. The outer beam particles first become paramagnetic and in doing so scrape off the edge of the beam and form a large minor radius hollow beamlet. As the energy continues to increase, the scrapeoff point moves inside the beam and inner beam particles continue to add to the outside of the hollow beamlet. The process is completed when the beam has turned itself completely inside out and has gone from a solid to a hollow beam.
Although this process is complicated, it is very easy to see that in making the transition, the beam must turn itself inside out. To show this, it is only necessary to invoke the conservation of toroidal canonical momentum P.sub.74 . If the poloidal magnetic field is given by .gradient..PSI..times.i.sub..theta. /R, then ##EQU12## To evaluate P.sub.74 , note that .gamma.=(E-q.phi.)/mc.sup.2 where .phi. is the electrostatic potential. For a cylindrical beam of radius r.sub.b, ##EQU13## The flux .PHI. has three components. First there is the flux of the vertical field itself, assumed uniform ##EQU14## Secondly, there is the flux associated with the focusing field. If the field index is 1/2, this is ##EQU15## Note that .PHI..sub.f has this form both for .rho.&lt;.rho..sub.b and .rho.&gt;.rho..sub.b. Finally, there is the flux associated with the self field, EQU .PHI..sub.s =(V.sub.74 /c.sup.2 R.theta.(.rho.) (14)
Thus if V.sub.74 .apprxeq.c, near the axis (.rho..apprxeq.0) one has that ##EQU16## Since qB.sub.z &lt;O for the modified betatron, one has the result that if n is large enough that the second term dominates (that is, if the beam is diamagnetic), P.sub..theta. (.rho.=0) is a relative minimum. However far from the beam P.sub..theta. is dominated by the focusing force which have the opposite sign. Thus P.sub..theta. as a function of .rho. for a diamagnetic beam is shown in FIG. 3a. On the other hand, if the beam is paramagnetic, the first term on the right hand side of Eq. (15) dominates so P.sub..theta. (.rho.=0) is a relative maximum, and P.sub..theta. (.rho.) is shown in FIG. 3b.
The crucial point is that in a configuration which has .theta. symmetry, P.sub..theta. is an exact constant of motion. Consider then the orbits at .rho.= 0 and .rho.=.rho..sub.b for a diamagnetic beam. The former is inside the latter and has a lower value of P.sub..theta. according to FIG. 1 a. After diamagnetic to paramagnetic tranistion, these values of P.sub..theta. cannot change. However a paramagnetic beam has the reference orbit at a relative maximum so that this must correspond to the orbit initially at .rho.=.rho..sub.b in the diamagnetic beam. Thus in making transition the beam must, at the very least, turn itself inside out.
Actually, as shown in Manheimer, supra; Finn, supra; and Grossman supra; not only does the beam turn itself inside out, it transitions from a solid to hollow beam. In doing so, the beam could strike the wall and thereby disrupt. Conditions for the beam to remain confined on transition are given in the three references. However, even if the beam does not remain initially confined, it is not certain it can remain confined long after suffering such a violent perturbation. If nothing else, the hollow profile produced is diocotron unstable.
One other potential difficulty with the modified betatron is the resistive wall instability. If a beam of density n and radius .rho..sub.b is centered in a cylindrical tube of radius a, the frequency of a perturbation at frequency varying like exp il.phi. is ##EQU17## where V.sub..phi. is the rotation frequency of the electrons ##EQU18## and .omega..sub.b is the frequency of rotation generated by the focusing fields, ##EQU19## The focusing fields produce a rotation in the negative direction. Since q&lt;o, the rotation of the beam itself is in the positive direction for the case of a vacuum beam, f=o. The sign of the frequency is such that as long as .omega.&gt;o, wall resistivity gives rise to growth of this mode. This can be understood by noting that since the beam has only negative charge, the net force from any perturbation must be outward. Thus, wall resistivity will cause the beam to spiral outward, corresponding to instability. Since the natural frequency of the l=1 mode is very low, the sign of this frequency can be changed by the focusing fields, thereby stabilizing this mode. The required condition for this is that the beam current, as defined above, be in the low or intermediate regime. However the l=2 mode has a significantly larger frequency so that in the vacuum modified betatron, it cannot be stabilized by the focusing fields.